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\begin{center}
 {\LARGE {\bf Materials Physics I}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 WS 2007/08\\
 \vspace{8pt}
 {\Large\bf Tutorial 4}
\end{center}

\section{Hall effect and magnetoresistance}
The Hall effect refers to the potential difference (Hall voltage)
on the opposite sides of an electrical conductor
through which an electric current is flowing,
created by a magnetic field applied perpendicular to the current.
Edwin Hall discovered this effect in 1879.

Consider the following scenario:
An electric field $E_x$ is applied to a wire extending in $x$-direction
and a current density $j_x$ is flowing in that wire.
There is a magnetic field $B$ pointing in the positive $z$-direction.
Electrons are deflected in the negative $y$-direction
due to the Lorentz force $F_L=-ev\times B$
until they run against the sides of the wire.
An electric field $E_y$ builds up opposing the Lorentz force
and thus preventing further electron accumulation at the sides.
The two quantities of interest are:
\begin{itemize}
 \item the magnetoresistance
       \[
       \rho(B) = \frac{E_x}{j_x} \textrm{ and}
       \]
 \item the Hall coefficient
       \[
       R_H(B) = \frac{E_y}{j_xB} \textrm{ .}
       \]
\end{itemize}
In this tutorial the treatment of the Hall problem is based on a simple
Drude model analysis.

First of all the effect of individual electron collisions can be expressed
by a frictional damping term in the equation of motion for the momentum 
per electron.
By inserting the forces acting on the elecron into the equation of motion
the expression for the Hall coefficient $R_H=-\frac{1}{ne}$ can be found.

\begin{enumerate}
 \item Recall the Drude model. 
       Given the momentum per electron $p(t)$ at time t
       calculate the momentum per electron $p(t+dt)$
       an infinitesimal time $dt$ later.
       {\bf Hint:} What is the probability of an electron taken at random at
       time $t$ to not suffer a collision before time $t+dt$?
       If not experiencing a collision it simply evolves under the influence
       of the force $f(t)$.
       Combine contributions of the order of $(dt)^2$ to the term
       $O(dt)^2$.
 \item Write down the equation of motion for the momentum per electron
       by dividing the above result by $dt$
       and taking the limit $dt\rightarrow 0$.
 \item Sketch a schematic view of Hall's experiment.
 \item Find an expression for the Hall coefficient.
       {\bf Hint:} Insert an appropriate force into the equation of motion
       for the momentum per electron.
       Consider the steady state and acquire the equations
       for the $x$ and $y$ component of the vector equation.
       To find an expression for the Hall coefficient use the second  equation
       and the fact that there must not be transverse current $j_y$
       while determining the Hall field.
 \item Calculate the Hall field for a rectangular wire
       (width: $l=15\, cm$, thickness: $d=4\, mm$) made out of silver
       if there is a current $I=200\, A$ flowing
       and a magnetic field $B=1.5\, Vs/m^2$ applied
       perpendicular to the current.
       Silver has the relative atomic mass $A_r=107.87$ and
       density $\rho=10.5\, g/cm^3$.
       Assume that there is one valence electron per atom.
       The atomic mass unit is $u=1.6605 \cdot 10^{-27} \, kg$
       and $e=1.602 \cdot 10^{-19} \, As$.
       {\bf Hint:} Start by calculating the Hall coefficient of silver first.
\end{enumerate}

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